Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini
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Transcript of Quantum Monte Carlo methods applied to ultracold gases Stefano Giorgini
Quantum Monte Carlo methodsapplied to ultracold gases
Stefano Giorgini
Istituto Nazionale per la Fisica della Materia Research and Development Center on
Bose-Einstein CondensationDipartimento di Fisica – Università di Trento
BEC CNR-INFM meeting 2-3 May 2006
QMC simulations have become an important tool in the study of dilute ultracold gases
• Critical phenomena
Shift of Tc in 3D Grüter et al. (´97), Holzmann and Krauth (´99), Kashurnikov et al. (´01)
Kosterlitz-Thouless Tc in 2D Prokof’ev et al. (´01)
• Low dimensions
Large scattering length in 1D and 2D Trento (´04 - ´05)
• Quantum phase transitions in optical latticesBose-Hubbard model in harmonic traps Batrouni et al. (´02)
• Strongly correlated fermions
BCS-BEC crossover Carlson et al. (´03), Trento (´04 - ´05)
Thermodynamics and Tc at unitarity Bulgac et al. (´06), Burovski et al. (´06)
Continuous-space QMC methods
Zero temperature• Solution of the many-body Schrödinger equation
Variational Monte Carlo Based on variational principle
energy upper bound
Diffusion Monte Carlo exact method for the ground state of Bose systems
Fixed-node Diffusion Monte Carlo (fermions and excited states) exact for a given nodal surface energy upper bound
Finite temperature• Partition function of quantum many-body system
Path Integral Monte Carlo exact method for Bose systems
function trial where TTT
TT HE
Low dimensions + large scattering length
1D Hamiltonian
if g1D large and negative (na1D<<1) metastable gas-like state of hard-rods of size a1D
N
i jiijD
iD zg
zmH
112
22
1 )(2
21
222
)1(1
6 D
HR
namn
NE
at na1D 0.35 the inverse compressibility vanishesgas-like state rapidly disappearsforming clusters
1
323
2
1
2
1 03.1122
aa
maa
mag DD
DD
g1D>0 Lieb-Liniger Hamiltonian (1963)
g1D<0 ground-state is a cluster state
(McGuire 1964)
Olshanii (1998)
Correlations are stronger than in the Tonks-Girardeau gas
(Super-Tonks regime)
Peak in staticstructure factor
Power-law decay in OBDM
Breathing mode inharmonic traps
mean field
TG
Equation of state of a 2D Bose gas
)/1ln(2
22
2
D
MF
nan
mNE
Universality and beyond mean-field effects
• hard disk• soft disk• zero-range
for zero-range potential mc2=0 at na2D
20.04onset of instability for cluster formation
BCS-BEC crossover in a Fermi gas at T=0
-1/kFa
BCSBEC
...)(
615
1281
18
52// 2/3
3 mFmFF
b akakNE
BEC regime: gas of molecules [mass 2m - density n/2 – scattering length am]
am=0.6 a (four-body calculation of Petrov et al.)am=0.62(1) a (best fit to FN-DMC)
Equation of state
beyond mean-field effects
confirmed by study of collective modes (Grimm)
Frequency of radial mode (Innsbruck)
Mean-field equation of state
QMCequation of state
Momentum distribution
Condensate fraction
JILA in traps
2/130 )(
38
1 mmann
Static structure factor (Trento + Paris ENS collaboration)
( can be measured in Bragg scattering experiments)
at large momentumtransfer
kF k 1/acrossover from S(k)=2 free moleculestoS(k)=1 free atoms
New projects:
• Unitary Fermi gas in an optical lattice (G. Astrakharchik + Barcelona)
d=1/q=/2 lattice spacing
Filling 1: one fermion of each spin component per site (Zürich)
Superfluid-insulator transition
single-band Hubbard Hamiltonian is inadequate
)(sin)(sin)(sin2
)( 22222
qzqyqxmq
sVext r
0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
superfluid fraction condensate fraction
s
S=1 S=20
• Bose gas at finite temperature (S. Pilati + Barcelona)
Equation of state and universality
T Tc T Tc
Pair-correlation function and bunching effect
Temperature dependence of condensate fraction and superfluid density
(+ N. Prokof’ev’s help on implemention of worm-algorithm)
T = 0.5 Tc